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Results

where n_{ij} is the number of i j transitions^{}directly calculated from the adhesion score sequence; p_{10} +^{}p_{11} = 1 and p_{00} + p_{01} = 1. By definition, p_{01} is the probability^{}of having adhesion (i.e., X_{i+1} = 1) under the condition that^{}the previous adhesion test is not successful (i.e., X_{i} = 0).^{}It is also given a short notation p by definition. p_{11} is the^{}probability of having adhesion in the next test if the previous^{}test also results in adhesion. It is also given a short notation^{}p + p by definition. p represents an increment (positive or^{}negative) in the probability of adhesion in the (i + 1)th test^{}due to the occurrence of adhesion in the ith test, which can^{}be thought of as a memory index. If the i.i.d. assumption holds,^{}p_{01} = p_{11} = p and p = 0, and we recover the Bernoulli sequence.^{}p can be estimated from the average adhesion frequency because^{}it also equals P_{a}. If the i.i.d. assumption is violated (p_{01}^{}p_{11} and p 0), p will not be equal to P_{a} (see Eq. 5 below).^{}
Direct calculations of p_{01} and p_{11} for the data in Fig. 2 D–F^{}show that their values are close to each other and to the averaged^{}adhesion probability, P_{a}, for LFA1/ICAM1 interaction, providing^{}preliminary validation for the i.i.d. assumption. By comparison,^{}transition 1 1 is more favorable than transition 0 1 for the^{}TCR/pMHC interaction, whereas the opposite seems true for the^{}Ccadherin interaction, indicating that the i.i.d. assumption^{}might be violated for these two cases.^{}
Closer inspection of the scaled adhesion scores in Fig. 2 D–F^{}reveals that they are clustered at different sizes. It seems^{}intuitive that, for a given "cluster size" m (i.e., m consecutive^{}adhesions), the number of times it appears in an adhesion score^{}sequence contains statistical information about that sequence.^{}Our intuition is supported by the data in Fig. 3, which shows^{}the cluster size distribution enumerated from the adhesion score^{}sequences in Fig. 2. Compared with the distribution for the^{}LFA1/ICAM1 interaction (Fig. 3A), the distribution for the^{}TCR/pMHC interaction (Fig. 3B) has more clusters of large size,^{}whereas the distribution for the Ccadherin interaction (Fig. 3C)^{}has more single adhesion events surrounded by noadhesion events.^{}
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To quantify the differences among the three cases in Fig. 3,^{}we derived a formula to express the number, M_{B}, of clusters^{}of size m expected in a Bernoulli sequence of length n and probability^{}p for the positive outcome in each test:
The first term in the upper branch on the righthand^{}side of Eq. 2 represents summation over the probabilities of^{}having a cluster of size m in all possible positions, i.e.,^{}for clusters starting at i = 2 to i = n – (m – 1).^{}Clusters starting from X_{1} or ending with X_{n} are accounted for^{}by the second term in the upper branch. The lower branch of^{}our formula accounts for the sequence of all 1s. It becomes^{}apparent from the above derivation that Eq. 2 assumes equal^{}probability for the cluster to take any position in the sequence.^{}This can be thought of as a stationary assumption.^{}
The total number of positive adhesion scores in the entire sequence^{}can be calculated by multiplying Eq. 2 by m and then summing^{}over m from 1 to n. It can be shown by direct calculation that^{}mM_{B}(m, n, p) = np. Here, np is the expected total number of^{}adhesion events. This outcome is predicted and shows that Eq. 2^{}is selfconsistent.^{}
M_{B} is plotted vs. P_{a} (= p) in Fig. 4A for n = 50 and for clusters^{}of sizes m = 1–4. The actual cluster size distributions^{}enumerated from measured adhesion score sequences (e.g., Fig. 3A)^{}should be realizations of the underlying stochastic process.^{}We used computer simulations to characterize the statistical^{}properties of this stochastic process. The mean ± SEM^{}of the number of clusters of size 1 from 3 (open triangles,^{}mimic experiments where three to five pairs of cells were usually^{}tested) and 50 (filled triangles, a good approximation to Eq. 2)^{}simulated Bernoulli sequences for several P_{a} values are shown^{}in Fig. 4A, which evidently agree well with M_{B}(1, 50, P_{a}) given^{}by Eq. 2.^{}
We next extend Eq. 2 to the case of a Markov sequence by including^{}a singlestep memory. The four conditional probabilities defined^{}in Eq. 1 form a onestep transition probability matrix [P] of^{}a stationary Markov sequence. Using Bayes' theorem for total^{}probability, the unconditional probabilities for the (i + 1)th^{}test are related to those for the ith test by [P]:
By applying the Chapman–Kolmogorov equation^{}(8), the nstep transition matrix can be obtained as [P^{(n)}]^{}= [P]^{n}. With the initial condition of P(X_{1} = 1) = p and P(X_{1}^{}= 0) = 1 – p, it can be shown by mathematical induction^{}that P(X_{i} = 1) = p(1 – p^{i})/(1 – p). The formula^{}for the expected cluster size distribution can now be extended^{}as follows:
Setting p = 0 reduces Eq. 4 to Eq. 2, as required. Eq. 4 has^{}another special case at p = 1 – p when it simplifies to^{}M_{M}(m, n, p, 1 – p) = p(1 – p)^{n}^{–m}, which describes^{}the situation where the first adhesion event in the sequence^{}would increase the probability for adhesion in the next test^{}to 1, leading to continuous adhesion for all tests until the^{}end of the experiment.^{}
Experimentally, the adhesion probability can be estimated from^{}the adhesion frequency F_{n}. The expected value of F_{n} can be calculated^{}as follows:
If p = 0 (i.e., a Bernoulli sequence), F_{n} approaches p as n^{}becomes large. However, if p 0 (i.e., a Markov sequence), then^{}F_{n} approaches p/(1 – p).^{}
M_{M} is plotted vs. P_{a} (related to p and p by Eq. 5) in Fig. 4B^{}for n = 50, m = 1 (i.e., solitary 1s bound by 0s from both ends)^{}for p ranging from –0.3 to 0.5 in increments of 0.1. The^{}case of p = 0 (i.e., Bernoulli sequence) is plotted as a solid^{}curve. Cases of p > 0 (i.e., memory with positive feedback)^{}are plotted as dotted curves. Cases of p < 0 (i.e., memory^{}with negative feedback) are plotted as dashed curves. Increasing^{}p shifts the curve downward toward a smaller number of isolated^{}1s in an adhesion score sequence.^{}
It can be shown by direct calculation that mM_{M}(m, n, p, p) = p[n – p(1 – p^{n})/(1^{}– p)]/(1 – p). Here, the lefthand side sums adhesion^{}events distributed in various clusters of different sizes, and^{}the righthand side is the expected number of adhesion events^{}in n repeated tests, nE(F_{n}) (cf. Eq. 5). This predicted result^{}confirms that Eq. 4 is selfconsistent.^{}
In Fig. 3, Eq. 4 was fit (curves) to the measured cluster size^{}distributions (bars) (see Materials and Methods for procedure^{}detail). Three different types of behaviors can be clearly discerned.^{}A memory index p 0 was returned from fitting the LFA1/ICAM1^{}data in Fig. 3A. Because of the limited amount of data, small^{}fluctuations of p from zero could be observed even for Bernoulli^{}sequences, as seen from computer simulations of a small number^{}of cell pairs. Fitting the TCR/pMHC data (Fig. 3B) returned^{}a positive p, indicating the presence of memory with positive^{}feedback, whereas fitting the Ccadherin data (Fig. 3C) returned^{}a negative p, indicating the presence of memory with negative^{}feedback. These results support our preliminary conclusions^{}based on the observations in Fig. 2 and preliminary analysis^{}using the p_{01}, p_{11}, and P_{a} comparison.^{}
The above fittings used the distribution of clusters of all^{}sizes (i.e., all m values) for a given P_{a} to evaluate p. However,^{}differences in the distribution of cluster sizes are dominated^{}by the difference in the expected number of clusters of size^{}1 (Fig. 3, comparing the first bar in each panel). Thus, the^{}number of clusters of size 1 in experimental adhesion score^{}sequences can be used as a simple indicator to evaluate the^{}memory effect.^{}
Analysis so far has used the raw data shown in Fig. 2, which^{}have similar P_{a} values. To obtain further support with 10x more^{}data, we varied contact times and/or receptor–ligand densities^{}to obtain different average adhesion frequencies, which also^{}allowed us to examine the potential effects of molecular densities^{}on p through P_{a}. For each system, the experimental number of^{}clusters of size 1, M_{exp}(1), for each P_{a} value was plotted in^{}Fig. 4B to compare with the theoretical curves, M_{M}[1, 50, p(50,^{}P_{a}, p), p]. It can be seen that the LFA1/ICAM1 data are scattered^{}evenly from both sides of the solid curve corresponding to p^{}= 0. By comparison, almost all of the TCR/pMHC data are below^{}the theoretical curve for p = 0, and most of the Ccadherin^{}data are above that curve. These results further support our^{}conclusions regarding three types of behaviors.^{}
The memory index p was obtained from fitting experimental cluster^{}size distribution with Eq. 4 and is plotted in Fig. 5(solid^{}bars) along with p estimated from direct calculation (using^{}Eq. 6 below) (open bars). Comparable results were obtained by^{}both approaches for all P_{a} values tested for all three systems^{}exhibiting qualitatively distinct behaviors. p values for the^{}LFA1/ICAM1 system are not statistically significantly different^{}from zero (P value 0.23) except in one instance (P value =^{}0.06). By comparison, a vast majority of the p values for the^{}TCR/pMHC system are statistically significantly greater than^{}zero, whereas half of the p values for the Ccadherin system^{}are statistically significantly less than zero and are marked^{}with asterisks on the top of corresponding solid bars to indicate^{}P value 0.05.^{}
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